In Canada, "route" is pronounced "root", so "rooter" would make sense. However, I only hear it pronounced as "row-ter" ("row" rhymes with "how"), and I usually follow the pronunciation of the day to avoid having to repeat myself.
I always say "Rowter". My friends and family take the mick out of it as they all say "Rooter". (We're from the UK and I guess "Rooter" is the typical way people say it in the UK and "Rowter" is the more US Term used).